Build a Mathematical Mind, Chapter by Chapter
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Would you like to be a proficient mathematician… without using numbers?
There is so much more to math than geometry and calculus! It is present in almost every life aspect, from improving your communication skills to how to fit your luggage into your car.
Did you always hate math because you couldn’t understand complex formulas?
Don’t let a few equations or a bad teacher deter you from building a mathematical mind. Learn the best cognitive tools to revolutionize the way you make sense of problems and persevere in solving them.
Boost your critical thinking and analytical skills.
Mathematical thinking involves analyzing data, patterns, and relationships and evaluating information and arguments, which can help improve critical thinking skills.
Adopt a mathematician’s mindset. Tinker, invent, make educated guesses, describe with precision, and use probability to your advantage.
Build a Mathematical Mind – Even If You Think You Can’t Have One is an action manual that will help you sharpen your everyday life skills such as:
- improving your logic,
- understanding how probability works,
- and making estimations.
This is a research-backed math manual you'll love to read. It contains examples for faster learning and greater everyday impact.
Hone your problem-solving skills and make better decisions.
Albert Rutherford is an internationally bestselling author whose writing derives from various sources, such as research, coaching, academic, and real-life experience.
Improve your communication skills.
Mathematical thinking involves clearly and concisely explaining ideas and solutions, which can improve how you communicate. With enhanced precision, you will have a keen attention to detail and the ability to be accurate in your thinking and talking.
Increase your confidence.
Developing mathematical thinking skills can increase your confidence and self-esteem, being able to solve difficult problems and understand complex ideas.
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Transcript
Hello, listeners.
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Speaker:And it goes on and on.
Speaker:Pure all right, that's enough to get to business.
Speaker:Today is the day for Albert Rutherford's chapter by chapter preview of his book, Build a Mathematical Mind.
Speaker:Do you want to improve your logic or understand how probability works?
Speaker:Or get better at making estimations?
Speaker:Maybe you'd like to hone your problem solving skills and make better decisions, or improve your communication skills.
Speaker:According to our author, all those can be improved in this book, Build a Mathematical Mind.
Speaker:Even if you think you can't have one.
Speaker:Let's get to it.
Speaker:Here's the chapter by chapter preview.
Speaker:Chapter one mathematical Habits of Mind.
Speaker:Ask any adult how they feel about math, and aside from a few math enthusiasts, you get a lot of lukewarm responses.
Speaker:People may say, I hated math or I was never good at math.
Speaker:I was much better at reading or art or music or writing or sports or It was just so boring.
Speaker:In high school, my friend and I passed notes the whole time.
Speaker:We have all sorts of reasons for disliking math.
Speaker:Maybe we were taught in a drill and kill method that bored us to tears.
Speaker:Maybe we tried to fit in with a certain crowd in high school by convincing ourselves that we didn't like math.
Speaker:Think of Lindsay Lohan's character in Mean Girls.
Speaker:Maybe we even liked math until we got to that infamous train problem in algebra class.
Speaker:Most of us probably think we aren't very good at math and may have started to believe we weren't math people sometime in grade school.
Speaker:But what is a math person?
Speaker:What if I told you that you could be a math person too?
Speaker:In fact, anyone could be a math person.
Speaker:This chapter will convince you that you can and should learn to think like a mathematician.
Speaker:The rest of the book will show you how many of us have an idea in our heads of what a math person is.
Speaker:Maybe it was the kid in class who raised his or her hand the fastest, or the one who always went up to the board to solve proofs in geometry.
Speaker:Maybe it was the middle school mathlete, or the student who took college level courses in high school.
Speaker:Sure, one or two of these people may have solved previously unsolved problems amazing and stunning the world's math community.
Speaker:The rest of them most likely didn't revolutionize the field of mathematics, but just enjoyed math during their school years and maybe beyond.
Speaker:So why did they enjoy math?
Speaker:What habits of mind brought them success in mathematics?
Speaker:These people knew how to think like a mathematician.
Speaker:Maybe they were born with the predilection toward logical thought.
Speaker:Maybe they were trained by talented teachers.
Speaker:Or maybe they just enjoyed mathematics enough when they were young.
Speaker:They trained their own brains.
Speaker:The point is, they learned how to think like a mathematician, and so can you.
Speaker:Despite what you may have thought in high school, mathematicians have a lot in common with artists, musicians and other creative thinkers.
Speaker:Mathematics is a creative field that involves visualizing finding patterns, asking what if and experimenting what you learned in school, memorizing your times tables, or following steps to solve an algebra problem has little to do with the creative thinking mathematicians do.
Speaker:Many mathematics educators have argued for reforming the way math is taught in school because it has so little to do with what math actually is.
Speaker:In 2009, math teacher Paul Lockhart wrote A Mathematician's Lament, a short book that's become a foundational piece for many seeking to reform mathematics education.
Speaker:In his Lament, Lockhart argues mathematics is an art form akin to music or painting, but it hasn't been recognized as such.
Speaker:He faults the educational system, writing in fact, if I had to design a mechanism for the express purpose of destroying a child's natural curiosity and love of pattern making chapter Two become a Pattern Detective have you noticed when the weather turns cool, leaves die and fall off of the trees?
Speaker:Or cats and some dogs like to lie in sunbeams?
Speaker:Of course you've noticed these things.
Speaker:Humans are born with the propensity to look for patterns.
Speaker:Young children notice when they cry, a grownup comes running to comfort them.
Speaker:Or if they draw on the walls, the grownup will get mad.
Speaker:We expect certain behaviors and phenomena because we have observed what usually happens.
Speaker:So what is a pattern?
Speaker:You might be having flashbacks to math class when your teacher asks you to extend a pattern to find an algebraic rule.
Speaker:Don't worry, we'll try that later.
Speaker:But patterns are much more than that.
Speaker:They are everywhere in nature.
Speaker:Look closely at a snail shell and you will see a pattern.
Speaker:Look at the way water ripples after a stone is thrown and you'll see a pattern.
Speaker:Mathematicians notice and study patterns like these to figure out why they exist and what they can tell us about our world.
Speaker:Sure, they also create patterns for people to examine in algebra class, but the study of patterns exists because people wanted to figure out the world around them.
Speaker:In other words, patterns are a natural phenomenon, and the first step to understanding them is to notice them.
Speaker:Humans are born with a propensity to look for structure.
Speaker:Neuroscientists have proven through functional MRIs that human brains naturally look for patterns in a sequence of items.
Speaker:In fact, there's a term apophenia for the human tendency to see patterns in meaningless data that may involve visual, auditory or other senses.
Speaker:Our desire to find patterns is so strong, we look for them when there aren't any.
Speaker:Extreme.
Speaker:Pattern seeking behavior is the hallmark of obsessive compulsive disorder, autism spectrum disorder, and other conditions.
Speaker:It's much more common to search for patterns obsessively, as people do with these conditions, than it is to not see patterns at all.
Speaker:What distinguishes us from most of the animal kingdom is the desire to find structure in the information coming our way, says Robert Barkman, professor emeritus of science and education at Springfield College in Massachusetts.
Speaker:The neocortex, which accounts for 80% of a human brain's weight and is found only in mammals, forms the neural networks responsible for patterns.
Speaker:Humans are so good at pattern recognition that computers have yet to outperform us in that regard.
Speaker:The desire and ability to find patterns is an innate skill identified in babies.
Speaker:One of the primary ways babies use their pattern sniffing skills is through language acquisition.
Speaker:Think about the way you speak.
Speaker:You don't pause between each word.
Speaker:So a newcomer to this world, a baby can recognize where one word ends and the next begins.
Speaker:Studies have shown that babies as young as eight months recognize patterns in speech, such as when certain sounds are grouped together, or those moments when we do pause between words.
Speaker:This pattern recognition helps them make sense of the sounds they hear, leading to the seemingly miraculous moment when they say their first word.
Speaker:You should be getting the sense by now that pattern sniffing means more than recognizing the patterns.
Speaker:In the tile of a 1950s bathroom, patterns can be visual, auditory, or the regular way.
Speaker:Chapter Three Probability and Experimentation if you've ever played a dice game or watched a movie that involves gambling, you've probably heard of Lucky Sevens.
Speaker:You've probably heard the phrase elsewhere too, since it seems to be everywhere.
Speaker:There's a movie called Lucky Seven.
Speaker:Numerous restaurants and bars are called Lucky Sevens.
Speaker:There's a Lucky Seven Casino and even a 2021 film challenge in Las Vegas called the Lucky Sevens Film Challenge.
Speaker:Seven feature films.
Speaker:Seven Days to Shoot, $7,000 budget.
Speaker:The ultimate film challenge.
Speaker:Are you in?
Speaker:It seems impossible not to have heard of Lucky Sevens.
Speaker:Do you ever stop to think about why sevens are supposed to be lucky?
Speaker:It's not random.
Speaker:The luck of the number seven can be explained by probability.
Speaker:Specifically for games involving dice, let's imagine you're playing a game a gambling game, perhaps, with each player rolling a standard pair of dice on their turn.
Speaker:The dice have six faces, meaning each die has numbers one through six exactly one time.
Speaker:Here's an exercise for you find a piece of paper or open a new document on your computer and list all the possible outcomes of rolling two dice.
Speaker:Try to work systematically so you don't miss any.
Speaker:For example, if die one lands on a one, die two could land on a 12345 or six.
Speaker:Record each of those sums separately.
Speaker:Since those are all possible roles.
Speaker:Once you've worked through the possibilities for die one, you don't need to do the same for die two.
Speaker:Since you've already accounted for each role when you recorded a roll of one of die one and three on die two, for example, it's the same as landing three on die two and one on die one.
Speaker:No need to count twice.
Speaker:You should have 36 possible outcomes.
Speaker:Now look at those outcomes the sums of the two dice.
Speaker:How many ways are there to roll as a sum of the two dice?
Speaker:A 10.
Speaker:How many ways to roll a two?
Speaker:One.
Speaker:A three?
Speaker:Two.
Speaker:Keep counting the possible outcomes until you get to seven.
Speaker:How many ways are there to roll a seven?
Speaker:That's right, six.
Speaker:There are more ways to roll a seven than there are to roll any other total.
Speaker:So are sevens lucky?
Speaker:No, they're just mathematically more probable.
Speaker:Add personal history, culture and superstition into the mix, and a person might have many reasons for believing sevens are lucky.
Speaker:But in the terms of dice, the luck of number seven can be explained by mathematics.
Speaker:The probability of something happening is always described as a number between zero and one.
Speaker:It's usually described as a fraction or a percent.
Speaker:Zero means absolutely no chance of it happening.
Speaker:What's the probability of rolling a zero when you roll two dice?
Speaker:Zero.
Speaker:You can't roll that.
Speaker:In the language of probability, one means certain to happen.
Speaker:It's rare that something has a probability of one, meaning it's guaranteed to happen.
Speaker:We could say the probability of rolling a sum between two and twelve when rolling two dice is one or 100%, because those are the only possible outcomes.
Speaker:But in real life, very few things are 100% guaranteed to happen.
Speaker:Between those two extremes are fractions that describe how likely a particular outcome is.
Speaker:Probability is calculated by figuring out the number of favorable outcomes.
Speaker:Chapter Four describing and Speaking in Mathematical Language as you've seen, mathematics is all around us.
Speaker:Mathematical phenomena exists in nature.
Speaker:If you take the time to notice and have a little knowledge about what you're looking for, you can use mathematics to understand the world around you.
Speaker:But there's still something that separates the average person from the mathematician.
Speaker:Knowing the language of mathematics.
Speaker:Perhaps the math taught in secondary schools is a language a second, 3rd or fourth language.
Speaker:For many students, all that time we spend solving equations and deciphering word problems is practice for understanding the mathematical phenomena around us.
Speaker:Let's look more closely at the language of mathematics.
Speaker:The first thing to understand is that the way we say numbers in English makes less sense than the way they're said in many other languages.
Speaker:If English isn't your first language, you may have struggled to pick up the counting system.
Speaker:Like the customary measurements we use in the US.
Speaker:Feet, inches, pounds, et cetera.
Speaker:Our names for numbers aren't entirely logical.
Speaker:When we count past ten, we have special numbers for eleven and twelve, and then the set of numbers 13 through 19 tells us how the numbers are composed.
Speaker:13 means three and 1017 means seven and ten.
Speaker:Once we get to 20, the order switches with the number of tens coming 1st.
Speaker:21 means two tens and 187 means eight, tens and seven ones.
Speaker:Romance languages, the most common of which are French, Spanish, Italian, Portuguese and Romanian, have a similarly irregular set of names for the numbers eleven through 15.
Speaker:But then switch to the ten and structure.
Speaker:17 in Spanish is the Aciete, literally ten and seven.
Speaker:In French, it's dicept ten seven in Chinese, Japanese and Korean languages, the structure for counting in teens is much more logical.
Speaker:Eleven is ten.
Speaker:112 is ten.
Speaker:Two all the way through to 1920 is 210.
Speaker:30 is 310.
Speaker:90 is 910.
Speaker:97 is 910.
Speaker:Seven.
Speaker:The way you say it tells you the value of the number.
Speaker:Think how much easier it would be for young English speaking children to learn to count if our counting system were this logical.
Speaker:The English language falls short compared to Japanese.
Speaker:Again, when we look at how we say larger numbers in English, the number four two seven, for example, comprises four hundred s, two tens and seven ones.
Speaker:Students spend a lot of time learning expanded form so they can understand the value of each number.
Speaker:In our base ten system, they learn to decompose four two seven into 400 plus 20 plus seven.
Speaker:You may remember your second grade teacher pointing to the two and asking what it means.
Speaker:If you said two, like many other children do, your teacher probably shot you an exasperated look, wondering why you didn't know that two actually means 20.
Speaker:In Japanese, the number four two seven is said as four hundreds, two tens, seven.
Speaker:No decomposing is required because the language conveys the value of each digit.
Speaker:Think how much easier it would be to learn to add and subtract if our language conveyed the value of each digit so clearly.
Speaker:As children get older and move beyond the study of place value, they spend many years learning the language of mathematics, particularly algebra.
Speaker:Were you confused the first time you saw chapter five?
Speaker:Tinkering, breaking it down and putting it back together?
Speaker:When you were a child, did you ever take something apart to see how it worked?
Speaker:Even something as simple as a pen or a mechanical pencil can fascinate a curious child or a board middle schooler.
Speaker:If you ever did this, you know the value of tinkering with things.
Speaker:Discoveries don't usually come to us as epiphanies, but rather as the result of exploration.
Speaker:Mathematicians wouldn't be able to make the discoveries they do without tinkering.
Speaker:Educators and psychologists have pointed out for nearly a century that people learn by experimenting and trying new things.
Speaker:We construct knowledge rather than having knowledge poured into our brains.
Speaker:Piaget, Dewey and Montessori contributed to the theory of constructivism, which states children learn by experiencing new things and incorporating them into their existing schema, thereby constructing understanding.
Speaker:Constructivism or experiential education is what mathematicians have long known.
Speaker:Phenomena need to be tinkered with, broken down, examined, and then put back together to create knowledge.
Speaker:Tinkering has become more popular in schools as educators realize it leads to better understanding than memorization does.
Speaker:Many schools now have maker spaces or Stem or steam classes built on the foundation of tinkering and experimentation.
Speaker:Let's look at a concrete example of how tinkering breaking things down into smaller pieces and putting them back together comes up.
Speaker:In math class, elementary school teachers talk about composing and decomposing numbers to help young children gain number sense and an understanding of place value.
Speaker:When adding 13 and eight, for example, first graders might learn to decompose the 13 into its pieces 110 and three ones.
Speaker:They might tinker with the eight, thinking about ways to decompose it.
Speaker:That would make the problem easier for them to solve.
Speaker:Six and two, five and three.
Speaker:They might realize that seven and one give them friendly numbers to work with.
Speaker:The three ones from the 13 can be combined with the seven ones to make a new ten.
Speaker:So 13 plus eight can be understood as 110 from the 13.
Speaker:Another ten, three plus seven and one one.
Speaker:In other words, 21.
Speaker:You might be thinking that's a ridiculously complicated way to add two small numbers.
Speaker:Nobody is arguing that it's the most efficient way to add.
Speaker:Rather, it's a step towards fluency.
Speaker:Educators know memorization without understanding rarely works.
Speaker:Being able to work through this process of decomposition demonstrates and builds fluency with numbers.
Speaker:Students who can break numbers apart like this, looking at their pieces and thinking strategically about how to combine them again, have a much higher chance of succeeding with more sophisticated mathematics.
Speaker:They have a stronger number sense and understanding of place value.
Speaker:Many concepts in math can be understood by breaking them down.
Speaker:Let's look at a concept that may have scared you in Algebra One the distance formula.
Speaker:Many students stumble through Algebra One, barely understanding what they're doing, memorizing whatever they need to in order to pass tests.
Speaker:The distance formula is one particularly nasty formula that students often balk at or cry over when their teachers tell them to memorize it.
Speaker:This formula tells us the distance between two points on the coordinate plane can be found this way you might be asked to use the distance formula to find the distance between points P and Q on the graph below.
Speaker:For example, chapter six inventing Understanding Algorithms and Using Them.
Speaker:You recall your teachers telling you, don't ask why invert and multiply or keep Change flip these are tricks for remembering how to defide fractions.
Speaker:Tricks aren't inherently bad, but they don't tell you how things work.
Speaker:They are memorization tricks that teachers often rely on, believing they're helping their students succeed on future tasks and tests.
Speaker:Tricks like these often obscure the understanding students need to learn and remember mathematics.
Speaker:There have been several wellknown articles and books in recent years that have circulated in the math education community advocating against tricks or artificial rules see Nix the Tricks and 13 Rules That Expire for More.
Speaker:While memorizing rules and tricks can obscure learning, building rules for yourself can make you more efficient.
Speaker:That's what algorithms and math are all about.
Speaker:If you have school age children, you may have found yourself baffled by their math homework.
Speaker:Parents often complain about the way math is taught now, with an emphasis on deconstructing numbers, building understanding, and solving problems in different ways.
Speaker:The Common Core Standards for Mathematics, published in 2012, have been a source of much contention.
Speaker:Parents and some educators have argued we need to return to the traditional way of teaching math, with an emphasis on memorizing and following algorithms.
Speaker:It worked for us, they often say, so why change it?
Speaker:The problem is, it didn't work for us.
Speaker:The American education system has long been known for producing unequal results with significant racial and economic disparities.
Speaker:The US.
Speaker:Has also lagged behind other wealthy nations in international tests of academic progress.
Speaker:If you're still not convinced, ask the average American on the street to divide two fractions.
Speaker:They'll likely try to recite some algorithm they memorized years ago, but will misapply it and get the wrong answer.
Speaker:And that's 6th grade math.
Speaker:Try throwing in a mixed number, and you'll have them stumped.
Speaker:The Common Core Standards attempt to bring equity to mathematics education and prepare American students for a future yet unknown.
Speaker:They're based on years of research on how people learn.
Speaker:They're based, in part, on the same habits of mind.
Speaker:This book strives to teach you the habits of mind that mathematicians have.
Speaker:So what is the role of algorithms in the Common Core?
Speaker:Or, more broadly, in the life of a mathematician?
Speaker:Let's start by defining algorithm.
Speaker:An algorithm is a procedure for getting a certain outcome.
Speaker:It's a set of steps to be followed that will always work.
Speaker:In math class, the standard algorithm refers to how you, your parents, or your grandparents most likely learn to do something.
Speaker:For multidigit multiplication, for example, the traditional algorithm looks like this.
Speaker:But an algorithm is much more than the traditional way you learn to do something.
Speaker:Remember, an algorithm can be any set of steps that reliably works.
Speaker:If a fourth grader has another method for multiplying multidigit numbers, that reliably works, it's an algorithm.
Speaker:When you balk at your child's math homework, you're likely seeing alternative algorithms you didn't know existed.
Speaker:The most important aspect of an algorithm is that the person who uses it understands it and can rely on it for solving that type of problem.
Speaker:It's a strategy for making work more efficient rather than trying to invent a new way each time.
Speaker:For example, a person can say, oh, I know how to do this.
Speaker:I follow these steps.
Speaker:Algorithms don't appear out of nowhere.
Speaker:Chapter Seven visualizing Externalizing the Internal if you've ever spent time with a gymnast, a concert pianist, or anyone else performing at any elite level, you may have seen them visualizing a performance, imagining every step of it, trying to picture themselves performing it perfectly.
Speaker:You may have wondered why they were doing this.
Speaker:Rather than spending time practicing leaps or scales, they know something.
Speaker:Mathematicians also know visualizing is a powerful tool.
Speaker:Mathematicians are skilled at visualizing.
Speaker:Einstein attributed his success to the skill.
Speaker:My particular skill does not lie in calculation, he wrote, but rather in visualizing effects, possibilities and consequences.
Speaker:It makes sense that visualizing holds such power for humans.
Speaker:Approximately 30% of the brains of primates, which includes humans, is used for visual processing.
Speaker:No wonder we are such visual creatures.
Speaker:Researchers have identified five aspects of visualizing internalizing identifying, comparing, connecting and sharing.
Speaker:We'll examine each of these and discuss how you can hone these skills to incorporate visualizing into your life.
Speaker:Internalizing involves making sense of something in your head.
Speaker:This is the first step to understanding a problem, particularly a complex one.
Speaker:Let's imagine you're trying to do something that challenges most people packing a car for a big trip.
Speaker:People who are good at fitting everything into the back of a car aren't magicians.
Speaker:They're just good at internalizing a spatial problem.
Speaker:When someone gets ready to pack a bunch of suitcases and bags into the trunk of a car, they need to spend time internalizing the problem first.
Speaker:They might ask themselves, how many large suitcases are there?
Speaker:What irregular objects do I need to get in?
Speaker:Are there pockets of space somewhere?
Speaker:Maybe under the back seats where certain items would fit?
Speaker:Where can I put the bag of fragile items so it's protected and not squished?
Speaker:The talented packer spends time picturing the answers to these questions and manipulating items in their head before packing the car.
Speaker:If you watch this person in action, you'll see they rarely have to pack and repack the car.
Speaker:They are strategic about what they put where, and they get everything in securely.
Speaker:This is because they spent time internalizing the problem and have a plan for how to solve it.
Speaker:The identifying stage of visualization involves identifying or creating an image or model that might help you young students learn to do this.
Speaker:To help them solve math problems, many teachers use a strategy called Read, Draw, write, which asks students to draw a model or picture to help them solve word problems.
Speaker:They are supposed to read the problem, draw a model, then write the answer in a sentence.
Speaker:The RDW strategy was not created to torture kids or the parents trying to help them with their homework.
Speaker:Rather, it's based on research about how visualization, particularly the act of drawing, creates a stronger understanding and memories of the problem.
Speaker:The process of creating a model leads to better understanding.
Speaker:Sometimes we're faced with problems that beg for a drawing to help us solve them.
Speaker:Imagine, for example, you're trying to figure out how to arrange seats at tables for a large party or banquet.
Speaker:You may want to sketch the tables and chairs to help you see the best arrangement.
Speaker:This is a problem that appears in a lot.
Speaker:Chapter Eight guessing Making Estimations the final habit of mind is one you use every day, whether you realize it or not.
Speaker:Making Estimations every time you go to the grocery store, unless you bring a calculator with you, you probably estimate your spending.
Speaker:When you need a tank of gas, you may estimate how much it's going to cost before leaving the house, you may grab a $20 bill estimating that anything you need money for will cost less than that.
Speaker:Before leaving for work in the morning, you estimate how much time you'll need to shower and get ready.
Speaker:Estimating, or making educated guesses about numbers, comes up all the time in everyday life.
Speaker:You may find some situations easier to estimate than others, and you may have friends that seem better than you at estimating.
Speaker:Estimating involves number sense, which is a person's ability to understand and manipulate quantities.
Speaker:Some of us have stronger number sense than others.
Speaker:Educators have realized how important number sense is as it forms the foundation not only for estimating, but for grasping numerical and spatial concepts easily.
Speaker:Much of math class now focuses on building students number sense, particularly in the youngest grades.
Speaker:If you've heard of students doing number talks in class, no, these students are building number sense.
Speaker:The stronger their number sense, the better their ability to estimate will be.
Speaker:Let's look at how you use estimation in your everyday life.
Speaker:You use it not just for costs, as the previous examples illustrated, but also for all kinds of quantitative scenarios.
Speaker:Imagine you're walking into a museum or large office building, and you see a big set of stairs ahead of you.
Speaker:Without realizing it, you mentally approximate how many steps there are and therefore how much effort you'll need to exert before weather deciding to look for the elevator.
Speaker:If there are only a few stairs in front of you, you'll probably take them.
Speaker:If it's a hefty flight, you may decide you'd be better off with the elevator.
Speaker:Now imagine you're about to go for a long drive, maybe a once yearly trip back home to see relatives.
Speaker:The map on your phone can tell you approximately how long it should take, let's say 5 hours.
Speaker:Because you've driven this route before, you know when and where you might encounter traffic.
Speaker:You might estimate an additional 2 hours.
Speaker:If you're driving during rush hour, maybe you'll have a toddler in your car and you'll tack on another hour for planned bathroom breaks.
Speaker:When you talk to your family on the phone the night before, you may tell them you plan to leave at eleven and get there at seven because you've estimated those additional 3 hours.
Speaker:This scenario demonstrates that personal experience also plays a role in estimating well.
Speaker:An experienced salesperson might estimate their profit on a new product before it's available to the public.
Speaker:A kindergarten teacher can predict how long it'll take 25 five year olds to walk from the playground back to their classroom.
Speaker:A good knitter can estimate how much longer it'll take to knit a pattern sweater than a simple hat.
Speaker:An experienced chef knows what a pinch of salt or approximately a teaspoon of something looks like.
Speaker:These estimates are based not just on knowledge of quantity, but on personal experience.
Speaker:While you can't feign experience you haven't had, you can improve your number sense.
Speaker:One of the first things young students learn is what many math programs called friendly numbers, or benchmark numbers.
Speaker:In our base ten system, friendly numbers are usually multiples of ten or 100.
Speaker:Friendly numbers are much easier.
Speaker:Chapter Nine How Mathematics Change the World if you're still feeling skeptical about the value that mathematics can bring to your life, perhaps a short lesson in history will help convince you how powerful it can be.
Speaker:History has a funny way of being rewritten and manipulated, depending on who's telling the story and what purpose they want the historical information to serve.
Speaker:We rarely hear about how mathematicians change the world, but it's true.
Speaker:There are numerous examples throughout history of the incredible contributions mathematicians made to civilization.
Speaker:Let's start with the ancient Sumerians, one of the earliest known civilizations in the world.
Speaker:Often credited with being the cradle of civilization, the Sumerians lived in Mesopotamia, in the area that is modern day Iraq, and flourished from approximately 5000 BCE to 2000 BCE.
Speaker:Sumerian civilization bloomed in part because humans learned to cultivate farmland, which led to increased food supply, enabling population growth and the establishment of large population centers.
Speaker:City States what does a growing population center need to amass wealth?
Speaker:Math.
Speaker:Specifically, the Sumerians needed a numbering system and basic calculations to help them keep track of land and taxes.
Speaker:Ancient clay tablets from the Sumerian city of Ur give us evidence of how King Shulgi, who ruled over Ur from approximately 2094 to 2046 BCE, created the first mathematical state.
Speaker:Shulge had hymns written about his prowess in nearly everything, and even declared himself a god during his reign, so we can't be sure how much of a mathematical genius he was.
Speaker:He did, however, standardize weights and measures a crucial step in keeping track of state finances.
Speaker:Can you imagine trying to rule an empire without standardized weights and measures?
Speaker:The Sumerians also created one of the first numbering systems, or one.
Speaker:Of the first we have evidence of thanks to those clay tablets.
Speaker:Look at the numbers one through 59 written in cuneiform.
Speaker:Practice your ability to notice patterns and examine the table for a minute.
Speaker:What do you see?
Speaker:Do you notice how the number eleven is the symbol for ten next to the symbol for one, and the pattern continues all the way through 59?
Speaker:It's similar to the way we write numbers using place value to help us write numbers greater than nine.
Speaker:One crucial difference between their system and ours is that once they got to 60, they restarted with the same symbol used for one and then would use place value to create higher numbers from there.
Speaker:Remnants of their base 60 system still exist today.
Speaker:For example, in how we tell time.
Speaker:There was a small problem in the Sumerian system.
Speaker:Though the number one and 60 were represented with the same symbol, there was no symbol for zero to indicate 60 was one group of 60 and nothing in the place to the right of it.
Speaker:In other words, without a zero, there was no way to indicate the difference between numbers that relied on position or place to be understood.
Speaker:At some point, perhaps in ancient India, or perhaps in multiple civilizations at different times, a symbol was used to represent nothing zero.
Speaker:In a position, it may seem trivial, but without the symbol for nothing, people's ability to indicate large quantities was limited once a symbol for zero came into existence.
Speaker:Our Capacity chapter Ten final Words you've learned the habits of mind that mathematicians rely on and that underline modern mathematics instruction.
Speaker:You've learned to sniff out patterns, understand and use probability, speak in the language of mathematics, tinker invent visualize, and make educated guesses.
Speaker:You've also learned, if you didn't know it before, that mathematics shaped our history and plays a role in nearly every aspect of life.
Speaker:One of the key differences between mathematicians and everybody else is that mathematicians don't give up when they face a mathematical challenge.
Speaker:In fact, they enjoy these challenges.
Speaker:They know learning comes from perseverance.
Speaker:Mistakes lead to greater knowledge, and even seemingly insurmountable problems may have solutions.
Speaker:In short, they believe in themselves and their problem solving abilities.
Speaker:The other thing that mathematicians understand is that math makes sense.
Speaker:It's not an obscure topic designed to torture students and stump grownups.
Speaker:Math is logical, and it can be understood by anyone who takes the time to try to understand it.
Speaker:Once you begin to unlock puzzles that seem impossible to you, you'll see that you can do it.
Speaker:You, too, can make sense of complex mathematics and begin to harness the power of mathematics in your life.
Speaker:Whether you're an artist, an engineer, a mechanic, a bartender, a professor, a teacher, or anything else, the mathematical habits of mind you learned in this book can help you, as they helped centuries of thinkers before.
Speaker:You try to keep them in the forefront of your mind and draw on the habits you need at different times.
Speaker:This is how to think like a mathematician be able to pick the right tool and discern the most efficient way to solve a problem.
Speaker:Respectfully, ARP you enjoyed this episode of Voiceover Work and audiobook sampler.
Speaker:Where do you listen?
Speaker:Be sure to join us next Monday for the chapter by chapter preview of this book albert Rutherford's Build a Mathematical Mindset.
Speaker:You to close out we have a quote from Barbara Streisand, whose birthday is today along with Joe Keary from Stranger Things, kelly Clarkson you know her?
Speaker:And Thomas Sanders.
